Abstract
We are developing an approach to teaching important proportionality-based concepts to first grade students in a way that supports students’ future progress in the domain. We consider the proportionality between magnitudes as a basic relationship behind multiple cases, usually described mathematically as ratio or rate. The core of our strategy is the modelling of a situation of proportionality and its transformations by creating a compound unit. The key action is coordinated measurement (co-measurement): students work in pairs, and each student is in charge of changing one of two magnitudes, while preserving proportionality. Based on our successful experiments in Grades 2–6, we incorporate shared responsibility work organization (“joint actions”) and the idea of compound unit into Davydov’s mathematics curriculum for the first grade. We built a new module based on Davydov’s idea about the role of rule-mediated counting by sets. In this paper, we present the results of our study, showing that first graders can learn the idea of compound unit and work with two magnitudes of different kinds while preserving proportionality.
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Notes
Several articles from this issue present Davydov’s original teaching strategy in more details.
In this case, “balanced” means that the vessel neither floats nor sinks in a given “sea”.
To the best of our knowledge, this book has not been translated into English. Below we translate some quotes from this book.
Many problems claim that they are dealing with multiplicative relationships, but actually they allow students to consider only one kind of magnitudes (see Problem 1).
In another (and also incorrect) additive strategy applied to the same problem, the student calculates the difference between the target number of mushrooms and the corresponding number in the exchange rule (10 − 2 = 8), and then adds the difference to the number of pine cones (5 + 8 = 13).
In this way, we extended their already interiorized mental action to its verbal form as these students were planning their actions and providing verbal guidance to their partners.
Four students could not be interviewed due to a schedule conflict.
References
Alexandrova, E. I. (2009). Matematika: Uchebnik dlya pervogo klassa nachalnoj shkoli (Sistema Elkonina-Davydova) [Mathematics: Textbook for 1st grade of elementary school (Elkonin D.B. - Davydov V.V. system)]. Moscow, Russia: Vita-Press (In Russian).
Bass, H. (2018). Quantities, numbers, number names and the real number line. In M.G. Bartolini Bussi & X.H. Sun (Eds.) Building the Foundation: Whole Numbers in the Primary Grades. The 23rd ICMI Study. Springer Open, Cham, p. 465–475.
Behr, M. J., Wachsmuth, I., Post, T. R., & Lesh, R. (1984). Order and equivalence: A clinical teaching experiment. Journal of Research in Mathematics Education, 15(5), 323–341.
Bobos, G., & Sierpinska, A. (2017). Measurement approach to teaching fractions: A design experiment in a pre-service course for elementary teachers. International Journal for Mathematics Teaching and Learning, 18(2).
Brousseau, G., Brousseau, N., & Warfield, V. (2013). Teaching fractions through situations: A fundamental experiment (Vol. 54). Springer Science & Business Media. Dordrecht
Confrey, J. (1995). Student voice in examining “splitting” as an approach to ratio, proportions and fractions. In Proceedings of the 19th Annual Meeting of the International Group for the Psychology of Mathematics Education. Recife, Brazil: Universidade Federal de Pernambuco.
Cortina, J. L., Visnovska, J., & Zuniga, C. (2014). Equipartition as a didactical obstacle in fraction instruction. Acta Didactica Universitatis Comenianae Mathematics, 14(1), 1–18.
Davydov, V.V. (2008). Problems of Developmental Instruction. Nova Science Publishers (Original work published in 1986).
Davydov, V. V., & Elkonin, D. B. (1966). Vozrastnye vozmozhnosti usvoeniia znanii (mladshie klassy shkoly). [Age capabilities in knowledge acquisition (early school grades)]. Moscow, Russia: “Prosveshchenije” (in Russian).
Davydov, V. V., & Tsvetkovich, Z. H. (1991). On the objective origin of the concept of fractions. Focus on Learning Problems in Mathematics, 13(1), 13–64.
Engeness, I. (2020a). Lecture 12. The conditions for the development of the properties of the action. The phases of the development of the action. Learning, Culture and Social Interaction, 100432. https://doi.org/10.1016/j.lcsi.2020.100432 In press.
Engeness, I. (2020b). Part 3: The study on the development of human mental activity: Lecture 10. The development of mental actions and the orienting basis of actions. Learning, Culture and Social Interaction, 100430. https://doi.org/10.1016/j.lcsi.2020.100430 In press.
Galperin, P. (1989). Mental actions as a basis for the formation of thoughts and images. Soviet Psychology, 27(3), 45–64.
Galperin, P., & Georgiev, L. S. (1969). The formation of elementary mathematical notions. Soviet studies in the psychology of learning and teaching mathematics, 1, 189–216.
Gorbov, S.F., Zaslavskij, V.M., Morozova, A.V. (2015). Deyatelnostnyi podhod k matematicheskomu obrazovaniju shkolnikov. [Activity approach to student’s Math Education] Moscow, Russia: Author’s club. (In Russian)
Hilton, A., Hilton, G., Dole, S., & Goos, M. (2013). Development and application of a two-tier diagnostic instrument to assess middle years students’ proportional reasoning. Mathematics Education Research Journal, 25, 523–545.
Inhelder, B., & Piaget, J. (1958). The growth of logical thinking from childhood to adolescence: An essay on the construction of formal operational structures (Vol. 22). Psychology Press.
Lamon, S. J. (2012). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers. New York, NY: Routledge.
Leontiev, A. N. (1978). Activity, consciousness, and personality. Englewood Cliffs, NJ: Prentice Hall (Original in Russian published in 1975).
Lobato, J., & Ellis, A. (2010). Developing Essential Understanding of Ratios, Proportions, and Proportional Reasoning for Teaching Mathematics: Grades 6–8. Reston: National Council of Teachers of Mathematics.
Mellone, M., Spadea, M., & Tortora, R. (2013). A story-telling approach to the introduction of the multiplicative structure at kindergarten. Didactica Mathematicae, 35.
Ni, Y., & Zhou, Y. D. (2005). Teaching and learning fraction and rational numbers: The origins and implications of whole number bias. Educational Psychologist, 40(1), 27–52.
Polotskaia, E. (2017). How the relational paradigm can transform the teaching and learning of mathematics: Experiment in Quebec. International Journal for Mathematics Teaching & Learning, 18(2).
Rubtsov, V. V. (1989). Organization of joint actions as a factor of child psychological development. International Journal of Educational Research, 13(6), 623–636.
Siegler, R. S., & Chen, Z. (2008). Differentiation and integration: Guiding principles for analyzing cognitive change. Developmental Science, 11(4), 433–453.
Simon, M. A., & Blume, G. W. (1994). Mathematical modeling as a component of understanding ratio-as-measure: A study of prospective elementary teachers. Journal of mathematical behaviour, 13(2), 183–197.
Simon, M. A., & Placa, N. (2012). Reasoning about intensive quantities in whole-number multiplication? A possible basis for ratio understanding. For the Learning of Mathematics, 32(2), 35–41.
Simon, M. A., Placa, N., Avitzur, A., & Kara, M. (2018). Promoting a concept of fraction-as-measure: A study of the learning through activity research program. The Journal of Mathematical Behavior, 52, 122–133.
Smith, C., Snir, J., & Grosslight, L. (1992). Using conceptual models to facilitate conceptual change: The case of weight-density differentiation. Cognition and Instruction, 9(3), 221–283.
van den Akker, J., Gravemeijer, K., McKenney, S., & Nieveen, N. (Eds.). (2006). Educational design research. New York: NY Routledge.
Venenciano, L., & Dougherty, B. (2014). Addressing priorities for elementary grades mathematics. For the Learning of Mathematics, 34(1), 18–24.
Vergnaud, G. (1988). Multiplicative structures. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (pp. 141–161). Reston, VA: National Council of Teachers of Mathematics.
Vygotsky, L. S. (1986). Thought and language. Cambridge, MA: MIT Press (Original in Russian published in 1934).
Vysotskaya, E., Lobanova, A., Rekhtman, I., & Yanishevskaya, M. (2017). Make it float! Teaching the concept of ratio through computer simulation. In EAPRIL 2016 Conference Proceedings, p. 297–312.
Vysotskaya, E., Rekhtman, I., Lobanova, A., & Yanishevskaya, M. (2015). From proportions to fractions. In EAPRIL 2014 Conference Proceedings, p. 42–53.
Watson, A., Jones, K., & Pratt, D. (2013). Key ideas in teaching mathematics. Oxford, UK: Oxford University Press.
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We are grateful to Elena Polotskaia and anonymous reviewers for their valuable suggestions and comments.
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Vysotskaya, E., Lobanova, A., Rekhtman, I. et al. The challenge of proportion: does it require rethinking of the measurement paradigm?. Educ Stud Math 106, 429–446 (2021). https://doi.org/10.1007/s10649-020-09987-8
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DOI: https://doi.org/10.1007/s10649-020-09987-8